Racah Polynomials: Hypergeometric Orthogonal Systems

• Racah Polynomials are finite hypergeometric orthogonal polynomials defined on a quadratic lattice that underpin numerous recoupling schemes in algebra and quantum theory.
• They exhibit a rich algebraic structure through Racah algebras and Leonard pairs, linking them deeply with representation theory and discrete systems.
• Their applications span discrete quantum mechanics, multivariate polynomial analysis, and extensions to q, exceptional, and elliptic settings.

Searching arXiv for recent and foundational papers on Racah polynomials to ground the encyclopedia entry. Racah polynomials are a finite family of terminating Saalschützian hypergeometric orthogonal polynomials on a quadratic lattice. In the standard hypergeometric form they are given by

$$R_n(X(x);\alpha,\beta,\gamma,\delta) ={}_4F_3\!\left(\begin{matrix} -n,\ n+\alpha+\beta+1,\ -x,\ x+\gamma+\delta+1\ \alpha+1,\ \beta+\delta+1,\ \gamma+1 \end{matrix};1\right), \qquad n=0,1,2,\dots,N,$$

with

$X(x)=x(x+\gamma+\delta+1),$

and one of the truncation conditions

$$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$

for a non-negative integer $N$ (Mishev, 2014). They belong to the classical Racah level of the Askey scheme and are among the most general finite-variable hypergeometric orthogonal polynomials; many other discrete orthogonal polynomial systems arise as degenerations or limits of Racah polynomials (Mishev, 2014).

1. Hypergeometric form and quadratic-lattice structure

The defining hypergeometric series is built from the generalized hypergeometric function

$${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$

where $(a)_n$ is the rising factorial (Mishev, 2014). In the Racah case the ${}_4F_3(1)$ is terminating and Saalschützian, and the natural variable is not $x$ itself but the quadratic lattice $X(x)$ (Mishev, 2014).

A standard univariate normalization used in multivariate and divided-difference treatments is

$$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$

which is of degree $X(x)=x(x+\gamma+\delta+1),$0 in the quadratic lattice

$X(x)=x(x+\gamma+\delta+1),$1

(Tcheutia et al., 2016). In discrete quantum-mechanical notation, ordinary Racah polynomials also appear as

$X(x)=x(x+\gamma+\delta+1),$2

with

$X(x)=x(x+\gamma+\delta+1),$3

and $X(x)=x(x+\gamma+\delta+1),$4 (Odake et al., 2011, Odake, 2018).

The classical difference-equation side is equally central. In the univariate quadratic-lattice formulation, Racah polynomials satisfy

$X(x)=x(x+\gamma+\delta+1),$5

with divided-difference and averaging operators

$X(x)=x(x+\gamma+\delta+1),$6

(Tcheutia et al., 2016). This quadratic-lattice formulation is the basis for multivariate generalizations and for the interpretation of Racah polynomials as bispectral objects.

2. Algebraic and representation-theoretic interpretations

The algebraic structure behind Racah polynomials is the Racah algebra, a quadratic algebra generated by two algebraically independent operators $X(x)=x(x+\gamma+\delta+1),$7 with commutator $X(x)=x(x+\gamma+\delta+1),$8 and defining relations

$X(x)=x(x+\gamma+\delta+1),$9

$$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$0

$$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$1

together with a Casimir operator

$$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$2

(Genest et al., 2013). In finite-dimensional irreducible modules, one may choose a basis in which $$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$3 is diagonal and $$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$4 is tridiagonal, and vice versa; thus $$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$5 form a Leonard pair, and the transition coefficients between the eigenbasis of $$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$6 and that of $$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$7 are exactly Racah polynomials (Genest et al., 2013).

This Leonard-pair viewpoint is sharpened by the $$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$8 Racah problem. For three mutually commuting copies of $$\alpha+1=-N \quad\text{or}\quad \beta+\delta+1=-N \quad\text{or}\quad \gamma+1=-N$$9, with intermediate Casimirs $N$0, setting

$N$1

realizes the Racah algebra, and the overlap coefficients between eigenbases of $N$2 and $N$3 are the Racah coefficients, or $N$4-symbols, hence Racah polynomials (Genest et al., 2013). The corresponding one-variable reduction yields explicit hypergeometric Sturm–Liouville operators $N$5 whose polynomial eigenfunctions are hypergeometric and whose overlap coefficients are the polynomial objects of Racah type (Genest et al., 2013).

The same recoupling picture extends to $N$6-type representation theory in a different guise. For Rankin–Cohen brackets, the coefficients in the recoupling identity

$N$7

are expressed in terms of the Racah values

$N$8

so the transition matrix between two coupling bases of a triple tensor product is again given by Racah polynomials (Labriet et al., 2023). This places the classical hypergeometric formulas and the recoupling-coefficient interpretation within a common algebraic framework.

A more recent extension is the meta Racah algebra, generated by $N$9 with

$${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$0

whose finite-dimensional representations yield both classical Racah polynomials and finite families of Racah-type biorthogonal rational functions as overlap coefficients (Crampé et al., 30 Mar 2026). In this setting the standard Racah family arises from overlaps between two ordinary eigenbases, while the rational extensions arise from overlaps involving a generalized eigenvalue problem (Crampé et al., 30 Mar 2026).

3. Recoupling problems, superintegrability, and multivariate Racah polynomials

For four copies of $${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$1, the recoupling scheme produces bivariate Racah polynomials. In the basis $${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$2 associated with the chain $${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$3, and the basis $${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$4 associated with $${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$5, the overlap coefficients are precisely bivariate Racah polynomials of Tratnik type (Post, 2015). The same problem is equivalent to the generic quantum superintegrable system on $${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$6, whose Hamiltonian is the total Casimir of four $${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$7 copies, and the overlap coefficients are the expansion coefficients between separated eigenbases in different spherical coordinate systems (Post, 2015).

An explicit bivariate Racah family used in the divided-difference treatment is

$${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$8

$${}_pF_q\!\left(\begin{matrix} a_1,a_2,\dots,a_p\ b_1,b_2,\dots,b_q\end{matrix};z\right) =\sum_{n=0}^\infty \frac{(a_1)_n(a_2)_n\cdots(a_p)_n}{n!(b_1)_n(b_2)_n\cdots(b_q)_n}z^n,$$9

with quadratic lattice variables

$(a)_n$0

(Tcheutia et al., 2016). These polynomials satisfy a fourth-order linear partial divided-difference equation with polynomial coefficients in $(a)_n$1 and $(a)_n$2,

$(a)_n$3

$(a)_n$4

which is the bivariate analogue of the univariate Racah divided-difference equation (Tcheutia et al., 2016). The same paper derives explicit matrix coefficients for the three-term recurrence relations satisfied by any bivariate orthogonal polynomial solution of that equation, presents the monic bivariate Racah family, and solves the connection problem between two different Tratnik-type families (Tcheutia et al., 2016).

Higher-rank algebraic generalizations are realized by the higher rank Racah algebra $(a)_n$5, the subalgebra of $(a)_n$6 generated by all intermediate Casimirs $(a)_n$7, $(a)_n$8 (Bie et al., 2018). In rank one this recovers the classical picture where $(a)_n$9 encodes the bispectrality of univariate Racah polynomials; in higher rank, connection coefficients between bases of Dunkl-harmonics diagonalizing different Abelian subalgebras are multivariate Racah polynomials, and the Geronimo–Iliev commuting Racah operators appear as the images of suitable Abelian subalgebras (Bie et al., 2018). The same paper identifies every generator of ${}_4F_3(1)$0 as a discrete operator acting on multivariate Racah polynomials after suitable changes of basis (Bie et al., 2018).

4. Special subclasses and exact identities

A particularly rigid finite subclass is obtained by the parameter choice

${}_4F_3(1)$1

for a positive integer ${}_4F_3(1)$2. Then the quadratic lattice reduces to

${}_4F_3(1)$3

and the corresponding Racah polynomials are

${}_4F_3(1)$4

(Mishev, 2014). These are exactly the polynomials appearing in the Kresch–Tamvakis conjecture

${}_4F_3(1)$5

for all ${}_4F_3(1)$6, ${}_4F_3(1)$7 (Mishev, 2014).

For this family, an inversion formula for the discrete transform

${}_4F_3(1)$8

yields the exact relation

${}_4F_3(1)$9

equivalently

$x$0

for $x$1 (Mishev, 2014). When $x$2, the factor $x$3 vanishes and one obtains the linear relations

$x$4

together with the equivalent Pochhammer form (Mishev, 2014).

The Leonard-pair viewpoint also yields nongeneric identifications between families. For dual Hahn polynomials $x$5, if

$x$6

then $x$7 is a Leonard pair, and the same polynomials become Racah polynomials with respect to the same discrete inner product (Huang, 4 Aug 2025). In that regime the same $x$8 admit the $x$9 representation

$X(x)$0

with

$X(x)$1

so the identification is not a limit transition but a second Leonard-pair structure on the same orthogonal family (Huang, 4 Aug 2025).

5. Exceptional, dual, and elliptic extensions

Exceptional Racah and $X(x)$2-Racah polynomials arise in the framework of discrete quantum mechanics with real shifts and shape invariance. For parameters

$X(x)$3

with

$X(x)$4

the ordinary Racah and $X(x)$5-Racah systems are deformed by a degree-$X(x)$6 polynomial $X(x)$7 built from twisted parameters (Odake et al., 2011). The exceptional family $X(x)$8 is defined by

$X(x)$9

$$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$0

has degree $$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$1, and begins at degree $$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$2 rather than $$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$3 (Odake et al., 2011). These families remain complete because they are the eigenbasis of a finite-dimensional Hermitian Hamiltonian, but they do not satisfy the ordinary three-term recurrence relation (Odake et al., 2011).

A different dualization procedure applies to the multi-indexed $$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$4-Racah polynomials. If

$$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$5

then the dual family satisfies an ordinary three-term recurrence

$$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$6

but also higher-order difference equations coming from the $$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$7-term recurrences of the original multi-indexed family (Odake, 2018). This makes the dual family an ordinary orthogonal family of Krall type and yields new exactly solvable discrete quantum systems with real shifts; these systems satisfy closure relations but are not shape invariant (Odake, 2018).

An elliptic extension is obtained from a finite discrete reduction of the difference Heun equation. The monic elliptic Racah polynomials are defined by the principal minors

$$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$8

where $$r_n(\alpha,\beta,\gamma,\delta;s) =(\alpha+1)_n(\beta+\delta+1)_n(\gamma+1)_n \,{}_4F_3\!\left( \begin{matrix} -n,\ n+\alpha+\beta+1,\ -s,\ s+\gamma+\delta+1\ \beta+1,\ \delta+\beta+1,\ \gamma+1 \end{matrix};1\right),$$9 is a finite tridiagonal matrix arising from the truncated elliptic Heun operator (Diejen et al., 2021). They satisfy the three-term recurrence

$X(x)=x(x+\gamma+\delta+1),$00

and explicit orthogonality relations on the finite spectral set $X(x)=x(x+\gamma+\delta+1),$01 (Diejen et al., 2021). In the trigonometric limit $X(x)=x(x+\gamma+\delta+1),$02, they reduce to the standard $X(x)=x(x+\gamma+\delta+1),$03-Racah polynomials of Askey and Wilson, so they sit naturally as an elliptic extension of the top finite discrete branch of the Askey scheme (Diejen et al., 2021).

6. $X(x)=x(x+\gamma+\delta+1),$04-Racah, braid, Bethe, and Griffiths-type extensions

The $X(x)=x(x+\gamma+\delta+1),$05-Racah family retains the Racah recoupling interpretation in quantum-group form. In the Racah problem for $X(x)=x(x+\gamma+\delta+1),$06, the two intermediate Casimirs $X(x)=x(x+\gamma+\delta+1),$07 and $X(x)=x(x+\gamma+\delta+1),$08 acting on a fixed multiplicity space $X(x)=x(x+\gamma+\delta+1),$09 form a Leonard pair, and the recoupling matrix has entries

$X(x)=x(x+\gamma+\delta+1),$10

equivalently a $X(x)=x(x+\gamma+\delta+1),$11 expression (Crampé et al., 2021). The same paper shows that these intermediate Casimirs are related by braid-group conjugation, and the recoupling matrix is proportional to the braid word

$X(x)=x(x+\gamma+\delta+1),$12

so the matrix entries of braid group representations are explicitly given in terms of $X(x)=x(x+\gamma+\delta+1),$13-Racah polynomials (Crampé et al., 2021).

A different algebraic realization arises from scalar products of Bethe states. For a Leonard pair of $X(x)=x(x+\gamma+\delta+1),$14-Racah type with eigenvalue sequences

$X(x)=x(x+\gamma+\delta+1),$15

the $X(x)=x(x+\gamma+\delta+1),$16-Racah polynomials

$X(x)=x(x+\gamma+\delta+1),$17

are expressed as exact ratios of scalar products of Bethe states, for example

$X(x)=x(x+\gamma+\delta+1),$18

and its homogeneous and inhomogeneous Bethe-state variants (Baseilhac et al., 2022). The standard recurrence, difference, and orthogonality relations then become reinterpretations of Leonard-pair and Bethe-state structures (Baseilhac et al., 2022).

Higher-dimensional $X(x)=x(x+\gamma+\delta+1),$19-Racah-type extensions appear in the bivariate Griffiths polynomials of $X(x)=x(x+\gamma+\delta+1),$20-Racah type,

$X(x)=x(x+\gamma+\delta+1),$21

where each $X(x)=x(x+\gamma+\delta+1),$22 is a normalized one-variable $X(x)=x(x+\gamma+\delta+1),$23-Racah polynomial and

$X(x)=x(x+\gamma+\delta+1),$24

(Crampe et al., 2024). These polynomials are built from three one-variable $X(x)=x(x+\gamma+\delta+1),$25-Racah blocks, are governed by the rank-2 Askey–Wilson algebra $X(x)=x(x+\gamma+\delta+1),$26, and are bispectral and biorthogonal rather than orthogonal (Crampe et al., 2024). In the limit $X(x)=x(x+\gamma+\delta+1),$27, they reduce to the earlier Griffiths polynomials of Racah type (Crampe et al., 2024).

Taken together, these developments show that Racah polynomials occupy a structurally privileged position: they are finite quadratic-lattice hypergeometric orthogonal polynomials; overlap coefficients for recoupling schemes; matrix elements of Leonard pairs and triples; eigenfunctions of difference and divided-difference operators; and starting points for multivariate, exceptional, dual, elliptic, quantum-group, and biorthogonal extensions (Genest et al., 2013, Bie et al., 2018, Diejen et al., 2021, Crampé et al., 30 Mar 2026).